Countable-dimensional real projective space

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Definition

As a projective space

This topological space is the projective space corresponding to a countable-dimensional vector space over the real numbers. Explicitly, denote by ${\mathbb{R}}^{\mathrm{\infty }}$ the space of all sequences of real numbers with at most finitely many nonzero entries, where the addition and scalar multiplication are coordinate-wise. This is a countable-dimensional vector space over $\mathbb{R}$. We thus have an action of the multiplicative group ${\mathbb{R}}^{*}$ on it by scalar multiplication. The quotient of ${\mathbb{R}}^{\mathrm{\infty }}\setminus \{0\}$ by this action, given the quotient topology, is defined as the countable-dimensional real projective space and is denoted as $\mathbb{R}{\mathbb{P}}^{\omega }$ or $\mathbb{R}{\mathbb{P}}^{\mathrm{\infty }}$.

As a quotient of a countable-dimensional sphere by antipode identification

Define $S^{\mathrm{\infty }}$, the countable-dimensional sphere, as follows: denote by ${\mathbb{R}}^{\mathrm{\infty }}$ the space of all sequences of real numbers with at most finitely many nonzero entries, where the addition and scalar multiplication are coordinate-wise. $S^{\mathrm{\infty }}$ is the subset of ${\mathbb{R}}^{\mathrm{\infty }}$ comprising those sequences whose sum of squares of entries is $1$.

$\mathbb{R}{\mathbb{P}}^{\mathrm{\infty }}$ is defined as the quotient of $S^{\mathrm{\infty }}$ by the following equivalence relation: any element is identified with its antipode, i.e., the element obtained by taking negatives of all the entries. This quotient is equipped with the quotient topology.

As a classifying space

Further information: Groupprops:Group cohomology of cyclic group:Z2 (on the Group Properties Wiki)

This topological space is a path-connected aspherical space and its fundamental group is cyclic of order two. This follows from the definition in terms of the countable-dimensional sphere (making the countable-dimensional sphere its double cover) and the fact that the countable-dimensional sphere is contractible

Thus, it can be viewed as a classifying space for cyclic group:Z2. In particular, this means that the topological homology and cohomology groups of this space are the same as the group homology and cohomology groups of cyclic group:Z2.

Algebraic topology

Homology

Further information: homology of countable-dimensional real projective space

Cohomology

Further information: cohomology of countable-dimensional real projective space

Homotopy

The zeroth homotopy is a one-point set, the first homotopy group is cyclic group:Z2, and all higher homotopy groups are zero (i.e., the space is an aspherical space). This follows because the universal cover, the countable-dimensional sphere, is a double cover and also the fact that countable-dimensional sphere is contractible.